Geodesic
Unique minimum semipaired dominating sets in trees
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 35-53

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Let G be a graph with vertex set V. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V ∖ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.
Keywords: paired-domination, semipaired domination number 
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Haynes, Teresa W.; Henning, Michael A. Unique minimum semipaired dominating sets in trees. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 35-53. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a2/